Theorem undif5 4483

Description: An equality involving class union and class difference. (Contributed by Thierry Arnoux, 26-Jun-2024.)

Assertion

Ref	Expression
undif5	⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 ∪ 𝐵) ∖ 𝐵) = 𝐴)

Proof of Theorem undif5

Step	Hyp	Ref	Expression
1		difun2 4479	. 2 ⊢ ((𝐴 ∪ 𝐵) ∖ 𝐵) = (𝐴 ∖ 𝐵)
2		disjdif2 4478	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∖ 𝐵) = 𝐴)
3	1, 2	eqtrid 2782	1 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 ∪ 𝐵) ∖ 𝐵) = 𝐴)

Syntax hints:   → wi 4   = wceq 1539   ∖ cdif 3944   ∪ cun 3945   ∩ cin 3946  ∅c0 4321

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-nul 4322

This theorem is referenced by:  fressupp  32177  sucdifsn2  37407